Probability Theory: The Logic of Science
by E. T. Jaynes
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Book Summary Information
Author: E. T. JaynesEditor: G. Larry Bretthorst
Edition: Hardcover
Audio: English (Unknown); English (Original Language); English (Published)
Published: 2003-06-09
ISBN: 0521592712
Number of pages: 758
Publisher: Cambridge University Press
Book Reviews of Probability Theory: The Logic of Science
Book Review: A brilliant logical foundation for physical science but...
Summary: 5 Stars
Jaynes has done a brilliant job of constructing a logical framework for incorporating and explicating the crucial differences between causal(physical)independence and logical(epistemological)independence that will enter into the assumptions underlying applications of the probability calculus(statistical inference)to scientific analysis in physical science(physics,engineering,chemistry,biology,etc.).Jaynes shows repeatedly how one goes about applying maximum entropy and/or Shannon's information approach to a host of problems in the physical sciences.Jaynes also incorporates valuable historical commentary on a host of individuals,from Laplace and Boole to Keynes,Ramsey,Fisher,Neyman and Pearson,Jeffreys,Savage,etc.For instance,on pp.564-65,he corrects the mythology and misdirected criticism directed at Laplace's calculation of the probability that the sun will rise tomorrow,given that it has already risen a certain number of times before,using the rule of succession.Laplace made it clear that he was not using any of the extensive ,relevant background knowledge ,available a priori, in his calculations.Thus,relative to the evidence specified only,there is nothing incorrect about the answer arrived at by Laplace.Jaynes,however, needed to add some chapters that would deal explicitly with social science and liberal arts,as well as disciplines like education,educational psychology,economics and business.I will spend the rest of the review commenting on the broader aspects of the work of John Maynard Keynes and Benoit Mandelbrot that Jaynes has appeared to have overlooked.First,Keynes intended that his interval estimate approach to probability,based on partial orders that do not satisfy the assumption of a sigma algebra,was applicable to all areas of life.Thus,Keynes obtains a general theory that is applicable everywhere.Nothing written by Keynes in the A Treatise on Probability(1921) contradicts the work of Jeffreys or Jaynes since the specific areas of science that Jeffreys and Jaynes seek to apply a logical approach to probability to satisfy the conditions need to specify a continuous mapping of the real numbers into each other.Single number probabilities under such a mapping require that a sigma algebra be specified in order to calculate the appropriate sums of unions and intersections.The basic data of physical and life science(molecules,cells,genes,chromosomes,atomic and subatomic partcles,electrons,etc.)is generally independent ,homogeneous, and invariant through time.When one turns to the other fields mentioned above,this is not the case.Keynes complemented his interval approximation approach to decision making by systematically constucting a conventional coefficient of risk and weight,c, that is able to deal with the nonlinear types of effects generated by the type of data available in the social sciences.Keynes's c coefficient equals p/(1+q)[2w/(1+w)].Define A to be an outcome.The decision maker maximizes cA,as opposed to the expected value rule,maximize pA or the expected utility rule,maximize pU(A),where U is a utility function and p is a probability(p+q=1).w represents the weight of the evidence.It measures the completeness of the actual and potential available ,relevant evidence upon which an estimate of probability will be based.It is an index that is normalized on the unit interval,0<=w<=1.Keynes's approach explains and gives solutions for all of the socalled paradoxes of decision theory.I now turn to the theoretical and empirical work of Benoit Mandelbrot.Based on a massive amount of data analysis from a number of different countries drawn from a number of different financial markets(cotton,commodity,stock,money,currency,bond),Mandelbrot has shown empirically that price movements in these markets demonstrate both long and short run dependence and discontinuity over time.Mandelbrot has incorporated variables representing these effects along with variables representing skewness and kurtosis into a generalized model that simplifies under special conditions to the normal distribution .Unfortunately,it is the normal probability distribution(and its relatives,the t,F,and chi square)that is used practically everywhere in the social sciences.Jaynes discussion of turbulence effects in his two page discourse on economics(7.21,pp.233-234)is suggestive that he also is somewhat aware that a different approach to analyzing social data that is nonhomogeneous and subject to abrupt and discontinuous change over time is needed in the social sciences.
Summary of Probability Theory: The Logic of Science
Going beyond the conventional mathematics of probability theory, this study views the subject in a wider context. It discusses new results, along with applications of probability theory to a variety of problems. The book contains many exercises and is suitable for use as a textbook on graduate-level courses involving data analysis. Aimed at readers already familiar with applied mathematics at an advanced undergraduate level or higher, it is of interest to scientists concerned with inference from incomplete information.Mathematics Books
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